During and after the Jin Dynasty there was a massive movement of population to the south of the Yangtse River for safety. The Jin Dynasty broke up into small kingdoms. This is called the Sixteen Kingdoms period (五 胡 十 六 国 (304 to 439) and overlaps the establishment of the five Northern and four Southern Dynasties (南 北 朝 386 to 589). The Southern Dynasties had their capitals south of the Yangtse River in Nanjing. The four Southern Dynasties were: Liú-Sòng 刘宋 420-479, Southern Qí 南齐 479-502, Liáng 南梁 502-557, and Chén 南陳 557-589.

Politically this was a time of civil war, attacks from external invasion, and political upheavals within the ruling families as each jostled for power. These factors weakened the ruling hierarchy and thus opened the culture to change. The scholars and bureaucrats gained power and filled the vacuum in many instances.

Socially, the massive movement to the south disrupted traditions and exposed people to new cultures. The invaders from the north brought new art forms and social traditions. In the south, the northerners moved into a part of China that spoke a different dialect and had distinct customs. Both Buddhism and Daoism increased their influence as the strong Confucian tradition waned and people began to look for other answers. The Magao grottoes at Dunhuang were begun in 366 as movement back and forth between western countries and China increased. With Buddhism came new approaches to many cultural ideas.

Economically while the break up of the traditional ruling classes meant that many great fortunes were lost, this meant that there was opportunity for others. The strangle hold on resources had been broken. The move to the south of the Yangtse was to the richest farmland in China. The increased population provided the labor for its development. This area called Jiangnan is still the richest area of China today, it includes Suzhou, Shanghai and Nanjing.

Throughout China, but especially in the Southern Dynasties, there were great advances in science, painting, calligraphy, poetry and technology. The free flow of ideas and the malleability of the social scene opened the culture to new possibilities.

Zu Chongzhi (祖冲之 Zǔ Chōngzhī 429-500 AD) and his family were part of and products of this turmoil. The family originally came from Beijing in Hebei province in the north. His great grandfather had been an official in the Eastern Jin Dynasty. His grandfather moved south with the great migration. In Nanjing (called Jiankang in those days), his grandfather, an engineer in charge of construction, and father, a Minister of Appointment or technical consultant, served the Liú-Sòng Dyansty (刘宋 420-479).

Zu Chongzhi received the finest education from his father and had the additional advantage of a surrounding culture of new ideas, changing traditions, and loose restrictions. He was sent to an academy as a boy under the emperor's patronage and then to Nanjing University to study. As a young man he was appointed by the emperor to a position at the Imperial Institute (Hualin Xueshen). He served both the Liú-Sòng Dyansty (刘宋 420-479) and the Southern Qí Dynasty (南齐 479-502).

While at the Imperial Institute, Zu Chongzhi worked on the problem of finding the value of pi π. He came very close, giving multiple solutions of differing exactness. He estimated it as falling between 3.1415926 and 3.1415927. He gave the fraction as ³⁵⁵⁄₁₁₃. He also suggested a second fraction ( ²²⁄₇) for rough approximations. This replaced Liu Hui's estimate of 3.141014 made in 263 AD and Zhang Heng's earlier estimate of π pi as the square root of 10 or 3.1622. Today the value of pi can be calculated to millions of decimal places, but here is a sample for comparison: 3.1415926535897932384626433832795028841971. (For those of you who have forgotten their grade school math, the area of a circle = πr² where r is the radius).

As Zu Chongzhi gained experience and showed his talents, the emperor appointed him to various posts in government both in the provinces and in Nanjing. Even though he had political and bureaucratic responsibilities he continued his own research and study. While working on the staff of the governor of present day Zhenjiang he started studying the Yuanjia calendar of 443. He worked out improvements to the calendar and presented them to the emperor in 462. This calendar was to become the Dà Míng (大明历) calendar and would include many of the discoveries he made in astronomy.

One of those measurements was Zu Chongzhi's calculation of the precession, suì chā (岁差 yearly difference), the movement of the earth's axis on a large circle in relation to the star field. The first measurement of precession had been made by an Eastern Jin astronomer Yu Xi (281-356 AD) as 1º each 50 years. It was important not only for the calculation of the calendar, but also for understanding star positions and navigation over time. Polaris will not always be the North Star. Below is a graph showing the precession as the North Pole sweeps along a circular arc pointing at different stars over time. Using a different method than Yu Xi, Zu Chongzhi measured the precession as covering 1º in 45 years and 11 months. This wasn't accurate, but served as the beginning for future revision by later astronomers. Today's measure is 1º each 71.6 years. Zu Chongzhi was the first to incorporate the measure into a calendar.

Having freed himself from the conventions of defining orbits, he went on to measure other orbits that had differing periods. He measure the sidereal year, the time it takes for earth to make one full orbit of the sun relative to the stars and the tropical year, the time it takes the earth to pass from one equinox point to the next.

In addition to astronomy, Zu Chongzhi made contributions to other fields. As a social engineer, he recommended that the emperor strengthen the country by improving farming and irrigation. He saw agriculture as the backbone of both the daily administration of the country and the military defense from invasion. As a mechanical engineer, in 478 he was asked by a military commander, Xiao Daocheng to recreate the legendary south facing chariot (Zhǐnán Chē 指南车) invented in 255 by Ma Jun (马钧) of the Wei Dynasty (220-265). The solution found by Ma Jun had been lost to history, so Zu Chongzhi had to reinvent the chariot.

Zu Chongzhi succeeded. Essentially, it is a cart that has a figure mounted on a freely moving stand such that no matter which direction the cart is traveling, the figure points to the south. Militarily this would be an advantage to Xiao Daocheng, but there is no record of it ever having been used in a campaign. A short time later, Xiao Daocheng usurped the throne and became emperor. Zu Chongzhi's design is also lost to history.

With his son Zǔ Gèng (祖暅), he wrote Zhuì Shù (綴术) The Method of Interpolation which became one of the classics of Chinese literature and science. It was required reading in the Imperial Academy of the Tang Dynasty, but over time it was dropped from the curriculum because it was too advanced for most of the students. No copies have been found as of yet.

He also worked with Zu Geng to calculate the area of a sphere. Zu Geng went on to make significant contributions to mathematics and astronomy in his own right. He developed a method for finding the diameter of a sphere when the volume is known. It is called "Zu Geng's Principle" or "Cavalieri's Principle" for an Italian mathematician who described it in the 1630's.

Zu Geng's son, Zu Hao, was also a mathematician and interested in refining the calendar. Unfortunately, the times caught up with him. The relative peace that had prevailed in the Northern and Southern Dynasties was wearing thin. As the different kingdoms gained wealth and power they increased their attacks on one another and there were increased attempts at coups. In 547, a general named Hou Jing, who had defected from the north and came to the Liang kingdom, decided he had a chance to take it over. He laid seige to the capital city of Jiankang and by 549 the Liang emperor and most of the populace starved to death. In the meantime, Zu Hao was in Guangling (Yangzhou) and remained loyal to the imperial line. He organized a counter attack in 550 but was quickly defeated and captured by Hou Jing. He was tied to a post and shot with arrows and then taken down and tied to five horse carts and pulled apart. Because they had supported Zu Hao, the people of the city were buried to their waists and then used for target practice by Hou Jing's archers. The Liang forces eventually captured and killed Hou Jing in 552, but by 557 the Liang Dynasty had been replaced by the Chen Dynasty.